Answer
1.consider the function f(x) = x^3 − 4x^2 + 2. calculate the limit of the difference quotient at x0 = 3 for f(x).
2.find the equation of a line passing through the same point that f(x) passes through at x = 3 with a slope equal to the limit you found in part a.
3.graph the function f(x) and the line you identified in part b. how do the graphs relate to each other? what does the slope of the line represent about the function at the given point?

Step-by-step explanation:

.

.

In other words, the graph of y = f(x) passes through the point (3, -7) where .

The point-slope form of a line in a cartesian plane is:

.

For this line,

is the point on the line, while

is the slope of the line.

The equation of this line will thus be

.

That's equivalent to

.

Refer to the diagram attached. The line touches the graph of y = f(x) at x = 3 without crossing it. The line here is thus a tangent to the graph of y = f(x) at x = 3. The slope of the line represents the instantaneous rate of increase of f(x) at .